Dear Adrian,
I think efficiency (c'*inv(X'X)*c)^-1) will be uninfluenced by the
mean of the regressors as long as the constant term is spanned by the
effective nuisance regressors, X*(I - c*pinv(c)) for a given contrast
c. E.g.
X1 = [kron(eye(3), ones(3,1)) rand(9,3)] % three groups, three covariates
D = eye(9) - ones(9)/9; % residual forming matrix for ones(9,1)
X2 = D*X1 % demeaned
ca = [0 0 0 1 0 0]'; % covariate
cb = [1 -1 0 0 0 0]'; % group difference
X1*(eye(6) - ca*pinv(ca)) % first three columns span constant
X1*(eye(6) - cb*pinv(cb)) % same, and same (span, not columns) for X2
inv(ca'*pinv(X1'*X1)*ca), inv(ca'*pinv(X2'*X2)*ca) % same efficiency
inv(cb'*pinv(X1'*X1)*cb), inv(cb'*pinv(X2'*X2)*cb) % same efficiency
I guess one answer might be to visualise efficiencies at the contrast
generation stage, instead of trying to represent the quality of the
design itself, which is perhaps not really meaningful -- even highly
correlated regressors are not necessarily a problem if you are looking
for a linear combination of them that is efficiently estimated (or of
course, a combination of other regressors, in which case two nuisance
regressors perfectly correlated with each other but not with the
contrast of interest will be no problem).
If we do want to show something before contrasts have been specified,
then I think you might be right that there is a case for showing
correlations rather than cosines... I wonder also if there might be a
case for showing the invariant part of the efficiency expression:
pinv(X'*X)... I just looked at this for X1 and X2 above, and the
problem is that they look very different for the first three group
regressors (because these span the constant mean which has been
removed from all regressors); so while it's obvious that ca is equally
efficient with the two different designs, it's not obvious that cb
is...
Since corrcoef(X1) and corrcoef(X2) are identical, that might not be a
bad compromise... Another idea might be to replace the redundant unity
diagonal of the correlation matrix with diag(pinv(X'*X)) (or perhaps
1./that), since that then relates to the efficiency of simple (single
1, rest zeros) contrasts over each column, in addition to showing the
correlation between other columns, which hints towards efficiency of
group comparisons, etc...
Interested to hear other opinions on this,
Ged
On 7 May 2010 16:12, Adrian Imfeld <[log in to unmask]> wrote:
> Dear Guillaume,
>
> Thanks for you reply. I agree that SPM is not displaying anything wrong in the orthogonality matrix. But I suspect that the orthogonality matrix is just not meaningful (or even misleading) for investigating the efficiency of the design. Wouldn't the correlation matrix of the regressors (invariant of mean regressor values) be more helpful? The recource you cite stresses the importance of the correlations between regressors for an efficient contrast calculation (section "Correlation between regressors"):
> http://imaging.mrc-cbu.cam.ac.uk/imaging/DesignEfficiency#head-525685650466f8a27531975efb2196bdc90fc419
>
> I still have to check whether the efficiency (c'*inv(X'X)*c)^-1) is influenced by the mean of the regressors or not...
>
> Thanks for your help, Adrian
>
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